The first circle in the paint stack will always be the easy one to hit. All that you have to do is place the paint splotches in the white space, and the entire circle is white space, so that's easy to do. Continuing forth, though, you'll have more challenging circles from circle number two on. So you'll have to do the timing properly.
When the circles get tougher, your best bet is not to shoot right away, but to watch the movement patterns. Wait for the circle to slow down, then to stop, then to get going again. Then shoot your colors onto the white spaces. Don't shoot too quickly or you'll lose quickly, as well.
As you complete levels, you'll get to unlock new color palettes for your circle stacks. Every one of them except for the boss battles has the same amount of colors available for you to use, and whenever you open chests, you have a chance to unlock new colors. When you do unlock them, choose them in the card menu, and it will affect all of your levels.
The exception to this is the boss levels, which happen every five levels. Bosses will not only have their own color palettes, but they will present a unique challenge, in that you will have something complete unique that you have to do in order to beat the level. One of the bosses, for example, will require you to place exactly one paint splotch per circle, but the rest of the circle will already be painted.
If you lose, you will be able to watch an advertisement video in order to continue on. You can do this as many times as you want per level, so if you want to beat the level, you can simply watch as many ad videos as you want whenever you lose, until you win at that level. There is no limit to the amount of ad videos that you watch, only how much patience you have for ad videos and how strong your internet connection is.